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Advanced control strategy of a solar domestic hot water systemwith a segmented auxiliary heater
T. Prud'homme*, D. GilletInstitut d'automatique, EÂ cole Polytechnique FeÂdeÂrale de Lausanne, CH-1015 Lausanne, Switzerland
Abstract
This paper describes improvements that can be introduced at both a structural and control level in order to improve the overall ef®ciency
of solar kits for domestic hot water supply.
The plant under consideration is a solar domestic hot water system (SDHWS) manufactured in Switzerland. The heat exchanger is a
mantle, which surrounds the entire storage tank.
One major structural improvement has been designed. It consists of the replacement of one single electrical element as an auxiliary heater
by three smaller ones with different lengths. This con®guration requires the manipulation of two additional actuators. An advanced control
strategy has been developed to handle the resulting complexity and to achieve higher performances. The actuators are the pump driving the
¯uid in the collector loop and the three electrical elements. A predictive control strategy has been proposed and validated. It requires
solving of an optimization problem and implementation of a state estimator.
Weather forecasts as well as prediction of the users' needs in terms of tapped water are required to implement this predictive control
strategy. The weather forecasts are provided on-line by the Swiss Meteorological Institute (SMI). The prediction of the users' needs is
updated every day using an extended Kalman ®lter.
Segmentation of the auxiliary heater coupled with a suitable advanced control strategy have led to signi®cant improvements in terms of
comfort and energy consumption. # 2001 Elsevier Science B.V. All rights reserved.
Keywords: Domestic hot water supply; Auxiliary heater; Solar kit
1. Introduction
The current control strategies in solar domestic hot water
system (SDHWS) are mainly based on manufacturers'
know-how. However, these strategies do not take into
account the evolution of the operational conditions, typically
the users' needs in terms of draw-off and the weather
conditions. They have been designed to work in the worst
case scenario, often leading to very conservative behavior.
The initial goal of this contribution was the development
of an advanced control strategy for all the variables manipu-
lated. At this time, no changes in the design of the SDHWS
were planned. However, previous results [1] have shown the
potential bene®t of implementing a segmented auxiliary
heater.
The SDHWS and its model are described in Section 2.
The segmentation of the auxiliary heater is detailed and
justi®ed in Section 3. The principle of the advanced control
strategy is presented in Section 4. The optimization proce-
dure and the state estimator are described in Sections 5 and
6, respectively. Section 7 presents how the predictions of
weather data and the predictions of the users' needs in terms
of draw-off are obtained. Simulation results and compar-
isons of the energy performance achieved with the advanced
control strategy proposed versus a conventional one are
given in Section 8.
2. The SDHWS and its dynamic model
A schematic view of the SDHWS under consideration is
given in Fig. 1.
In this ®gure, the location of eight available temperature
measurements Mi is indicated. Strati®cation in the storage
tank is well established owing to four valves controlled by an
autonomous switching law. Only one valve is opened at a
time. The ®rst valve (the highest one) is opened if
�M8 ÿM2� > 0, the second is opened if �M8 ÿM2� > 0
and �M8 ÿM3� > 0, etc. Basically, a valve is opened if
the temperature of the collecting ¯uid at the output of
Energy and Buildings 33 (2001) 463±475
* Corresponding author.
E-mail address: [email protected] (T. Prud'homme).
0378-7788/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 7 7 8 8 ( 0 0 ) 0 0 1 0 7 - 9
the collector is higher than the temperature in the storage
tank at the level of this valve, and if no upper valve can be
opened.
The storage, the mantle, the pipes of the collector loop and
the collector are modeled using 13, 10, 4 and 1 node(s),
respectively. In reality, the temperature of the water inside
the storage tank and inside the heat exchanger vary gradu-
ally. This simple model, although detailed enough, is in fact
well suited to develop and analyze control strategies without
cumbersome computational limitations.
The dynamic behavior of the SDHWS is de®ned by
computing the energy balance of each node. It leads to
one differential equation per node. Using the conditional
factor xi (which enables or disables the corresponding
term according to the node location), the energy balance
is represented by Eq. (1) for each node �s; i� of the store,
by Eq. (2) for each node �h; j� of the heat exchanger
and, by Eq. (3) for the node (c) of the collector and
by Eq. (4) for each node �p; k� of the pipes of the collector
loop.
Ms;iCpfdTs;i
dt� xiKh;s�Th;iÿ2 ÿ Ts;i� � _mLCpf�Ts;i�1 ÿ Ts;i�� �1ÿ xi�Ks;a�Tamb;in ÿ Ts;i� � _Qi (1)
Nomenclature
State variables of the SDHWS model
Tc temperature of the liquid inside the collector
(K)
Th;j temperature of the jth node of the heat
exchanger (K)
Tp;k temperature of the kth node of the pipe of the
collector loop (K)
Ts;i temperature of the ith node of the store (K)
Constant parameters of the SDHWS model
A collector area (m2)
c0 collector's optical efficiency
c1 collector's heat loss coefficient (kJ/(m2K))
Ccol thermal capacity of the collector (kJ/K)
Cpc specific thermal capacity of the collecting
fluid (kJ/(kgK))
Cpf specific thermal capacity of the fluid in the
store (kJ/(kgK))
Kh;a heat loss capacity rate from the heat exchanger
to ambient (kW/K)
Kh;s heat transfer capacity rate from the heat
exchanger to the store (kW/K)
Kp;a heat loss capacity rate from the pipe of the
collector loop to ambient (kW/K)
Ks;a heat loss capacity rate from the store to
ambient (kW/K)
Mh;j mass of the jth node of the heat exchanger (kg)
Mp;k mass of the kth node of the pipe of the
collector loop (kg)
Ms;i mass of the ith node of the store (kg)
Manipulated inputs
_mC mass flow rate of the collecting fluid (kg/s)
P1 power supply of the longest electrical element
(kW)
P2 power supply of the medium electrical ele-
ment (kW)
P3 power supply of the shortest electrical element
(kW). It should be noted that _Qi and Pj are
related. We haveP
all i_Qi � P1 � P2 � P3
_Qi auxiliary heater input of the ith node of the
store (kW)
Meteorological disturbances
IT global solar radiation on the collector surface
(kW/m2)
Tamb;ex ambient temperature around the collector (K)
Tamb;in ambient temperature around the store (K)
Other disturbances
_mL mass flow rate of the fluid in the store (kg/s)
Tin temperature of the liquid at the input of the
store (K)
Objective function
J objective function
Pelec power supplied to the three electrical elements
(kW)
Ppump power supplied to the pump driving the
collecting fluid (kW)
Psol power gathered by the collector (kW)
a trade-off factor between energy consumption
and comfort
Optimization procedure
a vector containing all of the decision variables
ai
a0 initial choice for a
a� optimal choice for a
gi gradient of J versus ai
G vector containing all of the gi
N number of decision variables
x vector containing the 28 temperatures of the
model
Energy balance
Eamb energy loss to ambient (kJ)
Eelec electrical energy consumed by the auxiliary
heaters (kJ)
Eload energy tapped by the users (kJ)
Esol solar energy gathered (kJ)
DE internal energy variation (kJ)
F1 solar fraction computed in the common way (%)
F2 solar fraction computed in the new way (%)
464 T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475
where
i � f1; . . . ; 13g; Ts;14 � Tin;
xi � 1 if i 2 f3; . . . ; 12g; otherwise xi � 0:
Mh;jCpcdTh;j
dt� Kh;s�Ts;j�2 ÿ Th;j� � _mCCpc�Ttest ÿ Th;j�� Kh;a�Tamb;ex ÿ Th;j� (2)
where
j � f1; . . . ; 10g
and
Ttest � Th;j; if the opened valve is below the jth node
Ttest � Tp;4; if the opened valve is at the same level as the
jth node
Ttest � Th;jÿ1; if the opened valve is above the jth node
CcoldTc
dt� c0AIT ÿ c1A�Tc ÿ Tamb;ex� � _mCCpc�Tp;2 ÿ Tc�
(3)
Mp;1CpcdTp;1
dt� _mCCpc�Th;10 ÿ Tp;1� � Kp;a�Tamb;in ÿ Tp;1�
Mp;2CpcdTp;2
dt� _mCCpc�Tp;1 ÿ Tp;2� � Kp;a�Tamb;ex ÿ Tp;2�
Mp;3CpcdTp;3
dt� _mCCpc�Tc ÿ Tp;3� � Kp;a�Tamb;ex ÿ Tp;3�
Mp;4CpcdTp;4
dt� _mCCpc�Tp;3 ÿ Tp;4� � Kp;a�Tamb;in ÿ Tp;4�
(4)
All the terms appearing in these equations are defined in the
nomenclature.
3. Segmentation of the auxiliary heater
The auxiliary heater introduced in every SDHWS is
essential to meet the requirements in terms of draw-off.
However, a bad control strategy of this auxiliary heater may
reduce the overall energy bene®t by destroying the strati®-
cation and disabling the two upper valves. This is particu-
larly true for the system considered, since its mantle heat
exchanger surrounds almost the entire surface of the
Fig. 1. The SDHWS and its division into nodes for modeling.
T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475 465
storage tank, including the part in which the auxiliary
heater stands. To overcome the drawbacks of such a struc-
ture, an improvement has been designed. This consists of
the replacement of the traditional single electrical element
by three smaller ones with different lengths and can be seen
from the schematic view in Fig. 1 as well as from the photo
in Fig. 2.
Together, these new elements must supply the same power
as the previous one, but may be activated independently
according to both the expected load and the solar radiation.
A combined auxiliary heater as proposed allows a better
control of the amount of heated water in the upper part of the
store and helps to preserve the strati®cation.
4. Predictive control principle
To achieve better overall performances, the controller has
to take advantage of additional information such as the
dynamic model of the system and the a priori knowledge
of the most signi®cant disturbances. In that way, the vari-
ables manipulated (inputs) can be adjusted according to a
predicted behavior of the system until satisfactory perfor-
mances are obtained. The resulting optimal inputs are then
applied to the real plant. The performance level is char-
acterized by an objective function that has to be minimized
using a suitable optimization algorithm. A scheme illustrat-
ing this principle is given in Fig. 3.
The term predictive results from the fact that changes in
the operational conditions are anticipated. For example, it
can be understood intuitively that the auxiliary heaters will
be turned on before each draw-off, thereby providing only
the amount of energy required to satisfy the users' needs. An
excellent review of this control strategy appears in [2]. Other
related works have been done in the area of predictive
control of passive and active solar systems [3±5].
As mentioned above, the performances desired are repre-
sented by an objective function. In this special case, the
objective is twofold: the electrical consumption is mini-
mized whereas the comfort of the users is maximized. The
objective function chosen J, as used to carry out this
optimization, is the following:
J �Z
one day
f�Pelec ÿ �Psol ÿ Ppump�� � a�Ts;1 ÿ Tset�2g dt
(5)
where Pelec is the electrical consumption of the three aux-
iliary heaters, Psol is the solar energy collected, Ppump the
power required by the pump to drive the fluid in the collector
loop and Ts;1 is the temperature in the upper part of the
storage tank. The parameter a is a trade-off factor and Tset is
the average temperature desired by the users. Tset has been
chosen equal to 55�C. To summarize, this control strategy
aims at minimizing the electrical consumption while keep-
ing the temperature of the water going out from the tank as
close as possible to the chosen Tset. This objective function is
minimized over a 1-day horizon. Ideally, it should be mini-
mized over the whole lifetime of the installation. However,
because of obvious computation limitations, and because of
the time-limited reliability of the weather forecasts, a 1-day
horizon has been chosen. This justifies the fact that Psol is
introduced in this objective function. Indeed, over a long
time, minimizing the electrical consumption or maximizing
the solar energy collected is strictly equivalent. However,
over the horizon chosen, they are not equivalent. Indeed, the
variation of the internal energy is not negligible in a 1-day
energy balance.
At this stage, it should be noted that no penalties have
been added in the cost function to take into account the fact
that electricity may be cheaper during day time than during
the night. This can be done easily by multiplying Pelec in the
cost function by a time-dependent term. In this paper, it is
assumed that the electrical elements can be turned on during
day time without any additional cost.
If P1, P2 and P3 are the power supply of the longest, the
medium and the shortest electrical elements, respectively,
Pelec is described as follows:
Pelec � P1 � P2 � P3 (6)
Fig. 2. The segmented auxiliary heater.
Fig. 3. Predictive control principle.
466 T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475
Pelec is also obviously linked to the term Qi that appears
in Eq. (1). The relation between these two terms is the
following:Xall i
_Qi � Pelec (7)
Psol is given by
Psol � _mCCpc�Tp;3 ÿ Tp;2� (8)
Ppump is given by
Ppump � K _m3C (9)
where K is a constant taking into account the pressure losses
in the collector loop and the efficiency of the pump.
The ¯ow rate in the collector loop and the power supply of
each electrical element are manipulated separately in order
to minimize the objective function.
The ¯ow rate can vary continuously between a lower and
an upper bound whereas the power supplies can, depending
on the con®guration chosen, either take only two discrete
values or also vary continuously between bounds. The
optimization has been carried out in both cases to ease
the comparison and justify the ®nal implementation scheme.
5. Optimization procedure
The optimization algorithm used to minimize the objec-
tive function is quite simple. First, the inputs to be optimized
are parameterized so as to transform the in®nite dimensional
problem into a ®nite dimensional one. There are two kinds of
parameterizations, depending on the type of inputs. If the
input can vary continuously, it is chosen piecewise contin-
uous with a 20 min sampling period. In that case, the values
of the input at each sampling time are optimized. If the input
can take only two discrete values, a ®xed sequence is chosen
and the switching times are optimized. These two parame-
terizations are summarized in Fig. 4. The arrows indicate in
which direction the curves are modi®ed during the optimi-
zation procedure.
Thus, all the inputs are described by a ®nite number of
variables called decision variables. These decision variables
are denoted ai and are grouped together in the vector a. a0
and a� represent the initial and the optimal choice, respec-
tively. The latter is the one minimizing the objective func-
tion.
For conciseness, the 28 differential equations that form
the dynamic model and the objective function given above
are written as follows:
_x � f �x; u; p; v� (10)
J �Z t0�D
t0
L�x; u; p; v� dt (11)
where x is the state vector containing the 28 temperatures
(state variables) of the model, its elements xi are the state
variables, u is the vector containing the four inputs manipu-
lated (flow rate in the collector loop and the power supply of
the three electrical elements), p the vector containing all the
constant parameters of the model and v the vector containing
all the disturbances (weather forecasts and users' needs in
terms of draw-off).
It is assumed that the optimization procedure is performed
at time t0 and covers a 1-day period (D). The vector p
containing the constant parameters of the model is known,
owing to an adequate identi®cation procedure. All the
disturbances contained in the vector v are estimated over
the horizon H � �t0; t0 � D�, the related estimation procedure
is described later. Thevalue of u over H and of x�t0� are needed
tosolve numerically theset ofdifferential equations (Eq. (10)).
Given the parameterization previously described, u is
completely de®ned by a. Therefore, given x�t0� and a set
Fig. 4. Alternative parameterizations of the inputs.
T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475 467
of decision variables a, x can be computed over H and, in
the same way, the objective function J can be evaluated.
Finally, if x�t0� is given, from an optimization point of
view, the objective function J can be seen simply as a
function of a.
The dimension of the vector a, which is also the number
of decision variables, is called N. An initial guess a0 for
a has to be chosen to start the optimization. As the optimiza-
tion problem is not convex, a0 has to be chosen carefully.
Then the algorithm computes a1, a2, . . . ; ak, which verify
J�a0� > J�a1� > J�a2� � � � > J�ak�. The procedure is
stopped when the difference J�ak�1� ÿ J�ak� becomes small
enough.
The algorithm used to perform this optimization is based
on the gradient G of the objective function J versus a. In the
present case, the elements gi of G cannot be computed
analytically. Thus, the perturbation method is chosen to
estimate them numerically:
gi � @J
ai
�ak� � J�ak � eei� ÿ J�ak�e
(12)
where e is a scalar chosen adequately to discriminate
between cost variation and numerical rounding errors,
and ei the ith vector of the canonical basis of RN .
The algorithm evaluates N � 1 times the objective func-
tion at each step, which means that the set of differential
equations of (10) has to be solved N � 1 times. Therefore, it
may induce a heavy computational load if the number of
decision variables is chosen inadequately.
To avoid the computation of unrealistic values for the
decision variables, such as a ¯ow rate that the pump cannot
afford, upper bounds ai and lower bounds ai of the decision
variables ai are also taken into account.
The way the algorithm implemented handles this type of
constraint follows. If the ith decision variable hits one of its
bound, then the ith element of the gradient of the objective
function is taken as equal to zero. In fact, it is a special case
of the projected gradient algorithm described in detail in [6].
If several variables hit their bound, the gradient is projected
to the intersection of the hyper-planes de®ned by the con-
straints. In our special case, it is the same approach as taking
the corresponding gradients of the objective function equal
to zero.
Once G has been computed, the correction of the decision
variables in the direction speci®ed by the gradient has to be
determined in order to reduce the objective function as much
as possible. This is a one-dimensional problem that can be
solved with the dichotomy, the Fibonacci or the golden
section search. These well-known methods are described
in [6]. The dichotomy method has proved to be suitable for
the SDHWS.
Despite the fact that the decision variables are determined
and can be applied over a 1-day horizon owing to the
approach previously described, the measurements available
can be used to regularly re®ne the optimum estimates. The
feedback mechanism resulting reduces the effect of model
mismatches and unforeseen disturbances. To implement this
feedback mechanism, a state estimator has to provide an
estimate x of x owing to the only eight measurements
available grouped together in y.
The complete feedback mechanism is schematized in
Fig. 5.
6. State estimator based on an extended Kalman filter
The estimator chosen in this work is an extended Kalman
®lter. As a preamble to its description, the discrete linear
Kalman ®lter is reviewed.
The dynamic behavior of the system for which the state
vector has to be estimated is described by Eq. (13). The
output equation is described by Eq. (14).
x�k � 1� � A�k�x�k� � w�k� (13)
y�k� � C�k�x�k� � z�k� (14)
where A�k� and C�k� are known matrices with appropriate
dimensions.
The noises w�k� and z�k� are independent, zero mean,
Gaussian with covariances given by Eqs. (15) and (16).
E�w�k�w�l�� � Q�k�dkl (15)
E�z�k�z�l�� � R�k�dkl (16)
for all k and l, where dkl is the Kronecker delta, which is 1 for
k � l and 0 otherwise.
Suppose that the initial state x0 is a Gaussian random
variable with mean x0 and covariance P0, independent of
w�k� and z�k�. Thus, it can be proven easily that x�k� and
y�k� are Gaussian random variables.
The ®ltering problem is to estimate x�k� using measure-
ments up to k. This reduces to the computation of the
sequence E�x�k�jy�0�; y�1�; . . . ; y�k�� for k � 0; 1; 2; . . ..This quantity is called xkjk. It implies the computation of
the quantity E�x�k�jy�0�; y�1�; . . . ; y�k ÿ 1�� for k � 0; 1; . . ..This quantity is denoted xkjkÿ1. The associated covariance
matrices Skjk and Skjkÿ1 must also be calculated.
xkjk is the conditional mean estimate, but it has been
proven that it is also the minimum variance estimate and
the conditional minimum variance estimate, at least with this
Fig. 5. Scheme of the feedback mechanism.
468 T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475
linear discrete system and with the assumptions on the
noises aforementioned.
The solution of this ®ltering problem is now given. The
proofs can be found in [7].
� Measurement-update equations:
xkjk � xkjkÿ1 � K�k��y�k� ÿ C�k�xkjkÿ1� (17)
K�k� � Skjkÿ1CT�k��C�k�Skjkÿ1CT�k� � R�k��ÿ1(18)
The matrix K�k� is called the gain of the Kalman filter.
Skjk � �I ÿ K�k�C�k��Skjkÿ1 (19)
� Time-update equations:
xk�1jk � A�k�xkjk (20)
Sk�1jk � A�k�SkjkAT�k� � Q�k� (21)
� Initial conditions:
xk0jk0ÿ1 � x0 (22)
Sk0jk0ÿ1 � P0 (23)
These equations can be implemented easily. The main
dif®culty related to this ®lter lies in the fact that a lot of
parameters must be chosen. The matrix R de®nes the quality
of the measurements. This matrix is often chosen diagonal.
If the ith measurement is good, the ith element of the
diagonal of R must be small. The Q matrix is also chosen
diagonal, each element of this diagonal de®nes the quality of
the corresponding differential equation. For example, if it is
clear that the ith equation is very close to reality then the ith
element of the diagonal of Q must be small. The opposite is
true. x0 is roughly evaluated owing to the measurements
taken at k0, and P0 is chosen diagonal. The elements of this
diagonal are the tuning parameters for the ®lter convergence.
As mentioned above, an extended version of the Kalman
®lter has to be used for the SDHWS. The differential
equations of the model of the system are neither linear
nor discrete. They are given by
_x�t� � f �x�t�; u�t�; p; v�t�� � w�t� (24)
The output equation is still linear and given by
y�tk� � Cx�tk� � z�tk� (25)
This equation shows that the measurements are taken at
discrete times tk, every 10 min in our special case.
The assumptions on the noises w and z are the same as in
the linear-discrete case.
This model must be linearized around a nominal trajec-
tory which is not known in advance. The solution is called
the prediction-correction method. The nominal state
x � x�tjtk� in the interval �tk; tk�1� is obtained by integrating
numerically the following system of non-linear differential
equations:
x�tjtk� � f �x�tjtk�; u�t�; p; v�t�� (26)
Thus, from tk to tk�1, starting from the initial condition
x�tkjtk�, this numerical integration provides x�tk�1; tk�. From
this, the nominal trajectory of the output is obtained easily
owing to
y�tk�1; tk� � Cx�tk�1; tk� (27)
Given the nominal trajectory, the Jacobian matrix can be
computed the following way:
F�tjtk� � F�x�tjtk�; u�t�; v�t�� � df �x�t�; u�t�; p; v�t��dx
����x�x�tjtk�
(28)
The equations of the extended Kalman filter are given by
� Measurement-update equations (correction):
K�tk� � S�tkjtkÿ1�CT�CS�tkjtkÿ1�CT � R�ÿ1(29)
x�tkjtk� � x�tkjtkÿ1� � K�tk��y�tk� ÿ Cx�tkjtkÿ1�� (30)
S�tkjtk� � �I ÿ K�tk�C�S�tkjtkÿ1� (31)
� Time-update equations (prediction):
_x � f �x�tjtk�; u�t�; p; v�t�� (32)
_S�tjtk� � F�x�tjtk�; u�t�; v�t��S�tjtk�� S�tjtk�FT�x�tjtk�; u�t�; v�t�� � Q (33)
� Initial conditions:
x�t1jt0� � x0 (34)
S�t1jt0� � P0 (35)
Time-update equations are numerically solved from tk to
tk�1. The parameters of this filter are the same as in the
linear-discrete case and are tuned in the same way.
The prediction is computed each time new measurements
are taken, every 10 min. The optimization procedure could
also be repeated every 10 min if there is enough time.
However, in our special case, the optimization is not per-
formed so often. The innovation sequence Inov�tk� is used to
decide whether a new optimization should be performed. It
is de®ned as follows:
Inov�tk� � x�tkjtk� ÿ x�tkjtkÿ1� (36)
Usually, all the elements of the vector Inov�tk� are small.
If one of them is bigger than a predefined constant, it
means that either the model has failed to predict the behavior
of the SDHWS or the disturbances (weather data or users'
needs in terms of draw-off) have been badly estimated. In
both cases, the optimization has to be performed to tackle
this problem.
In our special case, Inov�tk� is a 28-dimensional vector,
and its elements represent a difference of temperatures for a
given node of the model. Thus, the optimization is repeated
each time one of the element of Inov�tk� is greater than a few
degrees Kelvin (or Celsius).
T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475 469
7. Weather forecasts and prediction of the users' needs
It is obvious that the model is not suf®cient to predict the
behavior of the SDHWS. The solar radiation, the ambient
temperature around the collector and the users' needs in
terms of draw-off must be predicted. The temperature of the
water at the input of the storage tank as well as the ambient
temperature around the storage tank does not have to be
predicted. Their in¯uence is not signi®cant and they can be
chosen constant over H.
As for the weather forecasts, stochastic models can be
built [8]. However, the Swiss Meteorological Institute (SMI)
is able to provide via e-mail accurate 2-day forecasts. An
advanced model is used to compute these forecasts. It
describes all of the relevant atmospheric phenomena at
different scales, like storms, wind, rain, high clouds, snow,
etc. Obviously, we could not have afforded to create such a
huge model ourselves because of computational limitations.
With regards to the users' needs in terms of draw-off, the
tapped water predicted for each day of the week is the
average of the corresponding day over the last two months.
This prediction is updated every day at midnight.
The main problem is the fact that a ¯ow meter is too
expensive to consider for measuring the ¯ow rate of the
tapped water. Therefore, the extended Kalman ®lter des-
cribed in the previous section is also used to ful®ll this task.
For that purpose, the dimension of the state vector x is
increased by one. This new state vector is called xc. The new
element is the disturbance that has to be estimated. In this
case, it is the ¯ow rate of the tapped water _mL. Below, it
matches the ith disturbance vi. To apply the equations of the
extended Kalman ®lter, the dynamics of this disturbance vi is
de®ned as
dvi
dt�t� � fvi
vi�t� � nvi�t� (37)
Thus, this differential equation together with the 28 others of
the model of the SDHWS lead to the following general
representation:
_xc�t� � fc�xc�t; u�t�; p; v�t�� � wc�t� (38)
The dimension of the vector w has also been increased by
one. The 28 first elements of the vector wc correspond to the
elements of w. The last element is nvi. The assumptions on
this new element are the same as the others. The output
equation remains unchanged. With this new model, the
procedure used is exactly the same as the one described
for the extended Kalman filter in the previous section (just
replace x by xc and adapt the dimensions of the various
functions accordingly).
There are some new parameters that have to be de®ned to
make this ®lter work properly. First, the dynamics of the
estimated disturbance fvimust be de®ned. In fact, the
dynamics of this disturbance is not known in advance. It
is chosen to equal zero. However, owing to the noise nvi, it
can be indicated easily to the ®lter that this dynamics is not
known. This is done by choosing a high value for the cor-
responding term in the diagonal of the covariance matrix Q.
Finally, initial conditions for this disturbance vi and its
covariance are given. Both are taken as equal to zero.
A ¯ow meter has been installed in a pilot plant to compare
the ¯ow rate measured with the one estimated by the ®lter. A
typical result for this comparison is shown in Fig. 6. The
®lter has no a priori knowledge of the tapped-water pro®le
we have imposed. The pro®le chosen is standardized to
validate the SDHWS. Even if its shape may be different from
a measured one, it exhibits all the discontinuities necessary
to validate the ®lter.
Therefore, the users' needs in terms of draw-off are well
evaluated on-line owing to this ®lter. As mentioned above,
this estimation is used at the end of the day to update the
prediction used in the optimization procedure.
8. Control strategies validation
This section presents simulation results that illustrate the
performances of the various strategies applied to control a
modi®ed or a conventional SDHWS.
Comparison criteria have to be de®ned to assess the
energy performance improvement of this advanced control
strategy coupled to the segmented auxiliary heater. The
energy balance of the SDHWS is given by
Eelec � Esol � Eload � Eamb � DE (39)
with
Eelec �Z
B
Pelec dt �Z
B
�P1 � P2 � P3� dt (40)
Esol �Z
B
Psol dt �Z
B
_mCCpc�Tp;3 ÿ Tp;2� dt (41)
Eload �Z
B
_mLCpf�Ts;1 ÿ Tin� dt (42)
The horizon B over which these integrals are specified is the
horizon chosen to compare the two configurations.
Eelec is the electrical energy consumed and Esol is the solar
energy gathered. Eload, Eamb and DE are the energy tapped by
the users, the ambient energy loss and the variation of the
internal energy of the SDHWS, respectively. This last term
depends on the temperature variation of the 28 nodes over
the B horizon.
Two criteria F1 and F2 have been chosen. They are often
referred to as the solar fraction:
F1 � Esol
Eload � Eamb� Esol
Esol � Eelec ÿ DE(43)
F2 � Esol
Eload
� Esol
Esol � Eelec ÿ DE ÿ Eamb
(44)
F1 is obviously lower than F2. F1 penalizes the strategies
that lead to a great amount of ambient energy loss. There-
470 T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475
fore, even if F2 is used commonly, it seems that F1 is more
representative of the quality of a configuration.
The ®rst part of this section justi®es the choice of an
on±off power supply for the modi®ed SDHWS that
integrates a segmented auxiliary heater.
With this choice, the second part of the section quanti®es
the bene®t of such a modi®ed SDHWS compared to the
traditional one that integrates a unique auxiliary heater. The
energy performance improvements presented in this section
suppose that there are no model mismatches and that the
disturbances are perfectly predicted. In a real situation, these
conditions may not be well satis®ed. The implications of the
resulting discrepancies in the users' behavior and in the
weather forecasts will be discussed in Section 9.
8.1. Power supply alternative for the modified SDHWS
First, it is assumed that the electrical elements can be
supplied continuously. The long, the medium, and the short
electrical element can afford power supplies of 1.5, 1.0, and
0.5 kW, respectively. Thus, all of the inputs are chosen
piecewise continuous functions with a period equal to
two times the sampling rate. This leads to 288 decision
variables over a 1-day horizon (4 inputs times 72 decision
variables per input).
Curves in Fig. 7 exhibit the pro®les of the solar
radiation chosen to carry out the simulation and the corre-
sponding optimal ¯ow rate provided by the optimization
procedure.
Fig. 6. Comparison between measured and estimated tapped water.
Fig. 7. Solar radiation and optimal flow rate.
T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475 471
In Fig. 8, the pro®le of the water tapped and the corre-
sponding continuous optimal power supply of each electrical
element are shown. Owing to the predictive behavior of the
controller, the auxiliary heaters are used mainly just before
water is tapped. In this simulation, the amount of water
tapped is low. Thus, the longest electrical element is almost
never used. This proves the bene®t of the segmented aux-
iliary heater.
These results also suggest that it may not be necessary to
power continuously the auxiliary heaters. The shape of the
continuous optimal solution is close to an on±off sequence.
Thus, a sub-optimal solution can be obtained with less
computational load by looking for an on±off sequence with
a prede®ned number of switching times. The simulation has
been carried out and the results are shown in Fig. 9.
The general behavior is almost the same, the auxiliary
heaters are turned on just before the water is tapped and the
third electrical element is not used. All the elements of the
energy balance previously de®ned have been computed for
the 1-day period considered and can be seen in Table 1. In
terms of solar fraction, the performances are approximately
the same in both the continuous and on±off cases.
Temperature pro®les inside the storage tank are shown in
Fig. 10. The degree of comfort is very high in both cases,
since the temperature in the upper part of the storage tank
stays close to the set temperature Tset of 55�C.
8.2. Advanced control strategy versus conventional one
From now on, it is considered that an on±off power supply
of the auxiliary heaters is implemented.
The conventional strategy for the SDHWS under con-
sideration is the following:
P1 � 0
P2 � 0
P3 � 3:0 kW if�M1 �M2 �M3� < 55�CP3 � 0:0 kW if�M1 �M2 �M3� > 55�C
P1 and P2 are equal to zero because there is only the largest
electrical element in the conventional configuration. It can
afford a 3.0 kW power supply. It corresponds to the same
total power affordable as the configuration with the seg-
mented auxiliary heater. M1, M2 and M3 are the three
temperature measurements located in the upper part of
the storage tank (see Fig 1). The set temperature for this
conventional strategy has been chosen 55�C in order to ease
the comparison with our advanced control strategy and its
Tset, also equal to 55�C.
Fig. 8. Tapped water and continuous optimal power supply of each auxiliary heater.
Table 1
Comparison between continuous and on±off power supply in terms of
energy performances
Continuous
power supply
On±off power
supply
Variation
(%)
Eelec (kJ) 3.69e4 3.73e4 �1.08
Esol (kJ) 2.63e4 2.63e4 0.00
Eamb (kJ) 6.57e3 6.79e3 �3.3
DE (kJ) 1.53e3 1.71e3 ÿ11.76
Eload (kJ) 5.51e4 5.51e4 0.00
F1 (%) 42.64 42.49 �0.35
F2 (%) 47.73 47.73 0.00
472 T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475
The simulation has been carried out over a 28-day hor-
izon. It is enough to reduce the in¯uence of the variation of
the internal energy and also enough to cover a wide spectrum
of operational conditions (weather data and users' needs in
terms of draw-off). For conciseness, pro®les of the water
tapped and meteorological data are not given. They are
identical for the two con®gurations.
On the left-hand side of Fig. 11, the temperatures of the 13
nodes of the SDHWS model for the 28th day resulting from
the new con®guration are shown. On the right-hand side, the
temperatures for the same day resulting from the conven-
tional con®guration are shown.
All the elements of the energy balance have been com-
puted for the 28-day period and are shown in Table 2.
From the conventional con®guration compared to the new
one, Eelec has been reduced by 10.0%. This decrease is due to
both a decrease of Eamb and an increase of Esol. It results
from the fact that the mean temperature inside the storage
tank is lower in the new con®guration. This can be seen
clearly in Fig. 11. It should be noted that the mean
Fig. 9. Tapped water and on±off optimal power supply of each auxiliary heater.
Fig. 10. Left: continuous power supply; right: on±off power supply.
Table 2
Energy balances
Conventional
configuration
New
configuration
Variation
(%)
Eelec (kJ) 1.02e6 9.17e5 ÿ10.5
Esol (kJ) 7.36e5 7.76e5 �5.50
Eamb (kJ) 1.88e5 1.33e5 ÿ29.4
DE (kJ) 2.43e4 1.23e4 ÿ49.6
Eload (kJ) 1.55e6 1.55e6 0.00
F1 (%) 42.4 46.2 �8.94
F2 (%) 47.5 50.1 �5.47
T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475 473
temperature in the conventional con®guration is highly
dependent on the chosen Tset. The value chosen to carry
out these simulations is the same for the two con®gurations.
However, in the current practice, the value chosen is higher
than 55�C for the conventional control strategy. Indeed, the
temperature in the storage tank has to be high enough to
prevent the biggest peak of tapped water with the worst
weather. With the advanced control strategy proposed, a
lower Tset can be afforded because estimations of the users'
needs and forecasts of the most signi®cant weather data are
taken into account explicitly. Thus, it gives the anticipation
effect necessary allowing a lower mean temperature in the
storage tank.
9. Concluding remarks
An advanced control strategy coupled with a segmented
auxiliary heater have led to a signi®cant increase of the solar
fraction and a higher degree of comfort. Electrical elements
do not have to be continuously powered. Moreover, an on±
off control strategy is also more suitable from a computa-
tional point of view. Indeed, this optimization can be
achieved in a fraction of the 10 min sampling period.
However, as already mentioned, these are simulation
results. In practice, they would be dependent on the quality
of the disturbances' estimates. However, this problem is
partly tackled by the fact that the optimization is repeated
frequently. Indeed, each time new measurements are avail-
able, the state estimator computes new initial conditions for
the optimization algorithm which is therefore able to provide
updated optimal inputs. It will fail only in the case of real
disturbances very different from their estimates. In that case,
another degree of robustness is necessary. For example,
rather than keeping only the temperature of the highest
node in the storage tank close to Tset, it may be speci®ed
to the optimizer to keep the temperature of two or more
nodes in the storage tank close to this same Tset. It would
assure a given amount of energy in reserve in case of under-
estimated draw-off or over-estimated solar radiation. Thus,
the objective function would be:
J �Z
one day
f�Pelec ÿ �Psol ÿ Ppump�� � a1�Ts;1 ÿ Tset�2
� a2�Ts;2 ÿ Tset�2g dt (45)
To conclude, the control strategy has to be conservative
enough depending on the quality of the estimates. On one
hand, if the estimates are very good, the control strategy
would not be too conservative and the energy performances
will be excellent. On the other hand, if no a priori knowledge
of these disturbances is available, it will be difficult to get
better performances than the conventional control strategy.
As for commercial applications, it may be dif®cult to
consider getting information such as weather forecasts at
every SDHWS because it requires communication capabil-
ities. Moreover, the advanced control strategy is based on the
solving of a complex optimization problem that necessitates
signi®cant computation capabilities. However, there is no
doubt that these two points won't be problematic in the near
Fig. 11. Left: new configuration; right: conventional configuration
474 T. Prud'homme, D. Gillet / Energy and Buildings 33 (2001) 463±475
future and the energetic bene®ts induced by the new con-
®guration will fully justify the price of the modi®cations.
Acknowledgements
This work is supported by the Swiss Federal Of®ce of
Energy, Grant DIS 22260.
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